how to find adjoint of this operator on the space of C[0,1]?

Question

We are given $T$ is an operator on $C[0,1]$ as follows

$T(g(x))=\sum\limits_{k=1}^{m}p_kg(f_k(x)), p_k\in [0,1], f_k\in C[0,1]$, could anyone tell me how to show adjoint of this operator is as follows?

Answer 1

The dual of $C[0,1]$ is the space of all signed/complex measures on $[0,1]$ with total variation norm. we have $T^{*}(\mu)(g)=\mu (Tg)=\int Tg d\mu=\int \sum p_kg(f_k)d\mu=\sum p_k \int gd(\mu\circ f_k^{-1})$ and hence $T^{*}=\sum p_k \mu\circ f_k^{-1}$